Seminars and Colloquia by Series

Low-rank matrix completion for the Euclidean distance geometry problem and beyond

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 18, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rongjie Lai – Rensselaer Polytechnic Institute –
Abstract: The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. Instead of directly reconstruct the incomplete distance matrix, we consider a low-rank matrix completion method to reconstruct the associated Gram matrix with respect to a suitable basis. Computationally, simple and fast algorithms are designed to solve the proposed problem. Theoretically, the well known restricted isometry property (RIP) can not be satisfied in the scenario. Instead, a dual basis approach is considered to theoretically analyze the reconstruction problem. Furthermore, by introducing a new condition on the basis called the correlation condition, our theoretical analysis can be also extended to a more general setting to handle low-rank matrix completion problems under any given non-orthogonal basis. This new condition is polynomial time checkable and holds for many cases of deterministic basis where RIP might not hold or is NP-hard to verify. If time permits, I will also discuss a combination of low-rank matrix completion with geometric PDEs on point clouds to understanding manifold-structured data represented as incomplete inter-point distance data. This talk is based on:1. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, arXiv:1812.05786 2018. 2. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, accepted, IEEE. Transaction on Information Theory, 2018. 3. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017.

2019 Georgia Scientific Computing Symposium

Series
Applied and Computational Mathematics Seminar
Time
Saturday, February 16, 2019 - 21:30 for 8 hours (full day)
Location
Skiles 005
Speaker
Various speakers – GT, Emory, UGA and GSU

The Georgia Scientific Computing Symposium is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community.

This year, the symposium will be held on Saturday, February 16, 2019, at Georgia Institute of Technology. Please see

http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium

for more information

Convex-Nonconvex approach in segmentation and decomposition of scalar fields defined over triangulated surfaces

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 11, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin Huska – University of bologna, Italy
In this talk, we will discuss some advantages of using non-convex penalty functions in variational regularization problems and how to handle them using the so-called Convex-Nonconvex approach. In particular, TV-like non-convex penalty terms will be presented for the problems in segmentation and additive decomposition of scalar functions defined over a 2-manifold embedded in \R^3. The parametrized regularization terms are equipped by a free scalar parameter that allows to tune their degree of non-convexity. Appropriate numerical schemes based on the Alternating Directions Methods of Multipliers procedure are proposed to solve the optimization problems.

Convex Relaxation for Multimarginal Optimal Transport in Density Functional Theory

Series
Applied and Computational Mathematics Seminar
Time
Friday, February 8, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Lexing Ying – Stanford University

Please Note: We will go to lunch together after the talk with the graduate students.

We introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.

Global Convergence of Neuron Birth-Death Dynamics

Series
Applied and Computational Mathematics Seminar
Time
Wednesday, February 6, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joan Bruna Estrach – New York University –
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this talk, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated. Joint work with G. Rotskoff (NYU), S. Jelassi (Princeton) and E. Vanden-Eijnden (NYU).

An Adaptive Sampling Approach for Surrogate Modeling of Expensive Computer Experiments

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashwin Renganathan – GT AE

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary.

GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

An Adaptive Sampling Approach for Surrogate Modeling of Expensive Computer Experiments

Series
Applied and Computational Mathematics Seminar
Time
Monday, February 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ashwin Renganathan – GT AE

In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary

GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.

Synchronization of pendulum clocks and metronomes

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 14, 2019 - 01:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Guillermo Goldsztein – GT School of Math –
In 1665, Huygens discovered that, when two pendulum clocks hanged from a same wooden beam supported by two chairs, they synchronize in anti-phase mode. On the other hand, metronomes synchronize in-phase when oscillating on top of the same movable surface. In this talk, I will describe and analyze a model to help understand the conditions that lead to anti-phase synchronization vs. the conditions that lead to in-phase synchronization.

Finite Dimensional Balian-Low Theorems

Series
Applied and Computational Mathematics Seminar
Time
Monday, January 7, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 154
Speaker
Dr. Michael Northington – GT Math
Gabor systems, or collections of translations and modulations of a window function, are often used for time-frequency analysis of signals. The Balian-Low Theorem and its generalizations say that if a Gabor system obeys certain spanning and independence properties in L^2(R), then the window function of such a system cannot be well localized in both time and frequency. Recently, Shahaf Nitzan and Jan—Fredrik Olsen show that similar behavior extends to Gabor systems of finite length signals in l^2(Z_d). In this talk, I will discuss these finite dimensional results as well as recent extensions proven in collaboration with Josiah Park.

Nonparametric inference of interaction laws in particles/agent systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, December 3, 2018 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fei Lu – Johns Hopkins University –
Self-interacting systems of particles/agents arise in many areas of science, such as particle systems in physics, flocking and swarming models in biology, and opinion dynamics in social science. An interesting question is to learn the laws of interaction between the particles/agents from data consisting of trajectories. In the case of distance-based interaction laws, we present efficient regression algorithms to estimate the interaction kernels, and we develop a nonparametric statistic learning theory addressing learnability, consistency and optimal rate of convergence of the estimators. Especially, we show that despite the high-dimensionality of the systems, optimal learning rates can still be achieved.

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